I found someone who shared a formula that would produce permutations of a factorial up to 10 without VBA. Fortunately I just needed it to go to 9. That's 362,880 permutations. Once I had that, I produced all the sums (answers) for my Circuit Board Square (there are 8 sums per puzzle). I then concatenated those as a single string and did a =countif across all those entires.
This =countif calculation took ~3 hours to complete since it had to do it against all 362,880 entires. Once that was done I was able to see how many unique entires there were with these permutations for solving the puzzle. Out of 362,880 permutations, there were 213,248 entries with a count of 1.
It was important for me to determine these individual permutations because I'd hate to be in a position where I randomly inserted numbers, got sums, and realized there were multiple answers. An answer key that had a different answer than yours, which may also be valid, is a bad puzzle experience.
Now I have 213,248 puzzles to choose from. I think that's pretty good for a 100 page starter book!